Question: Simplify and expand the following expression: $ \dfrac{4y + 7}{y + 9}+\dfrac{4y}{5y + 1} $
In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(y + 9)(5y + 1)$ Multiply the first term by $\dfrac{5y + 1}{5y + 1}$ $ \begin{align*} \dfrac{4y + 7}{y + 9} \times \dfrac{5y + 1}{5y + 1} & = \dfrac{(4y + 7)(5y + 1)}{(y + 9)(5y + 1)} \\ & = \dfrac{20y^2 + 39y + 7}{(y + 9)(5y + 1)}\end{align*} $ Multiply the second term by $\dfrac{y + 9}{y + 9}$ $ \begin{align*} \dfrac{4y}{5y + 1} \times \dfrac{y + 9}{y + 9} & = \dfrac{(4y)(y + 9)}{(5y + 1)(y + 9)} \\ & = \dfrac{4y^2 + 36y}{(5y + 1)(y + 9)}\end{align*} $ Now we have: $ = \dfrac{20y^2 + 39y + 7}{(y + 9)(5y + 1)} + \dfrac{4y^2 + 36y}{(5y + 1)(y + 9)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{20y^2 + 39y + 7 + 4y^2 + 36y}{(y + 9)(5y + 1)} $ $ = \dfrac{24y^2 + 75y + 7}{(y + 9)(5y + 1)}$ Expand the denominator: $ = \dfrac{24y^2 + 75y + 7}{5y^2 + 46y + 9}$